We perform numerical experiments on Hamiltonian systems with three degrees of freedom. The correlation dimension, the maximal Lyapunov exponent, and an indicator of the deviation from the harmonic situation are measured. Our results indicate the existence of structures with noninteger correlation dimension at least for finite, but long, observational times. The transition from the motion on the Kolmogorov-Arnol’d-Moser tori to an ergodic motion on the constant-energy surface seems to occur smoothly as energy increases and is difficult to explain in terms of perturbative methods.

By employing the virial theorem and Hellmann-Feynman relation in combination with the pertinent uncertainty-principle inequality, one can derive a rigorous lower bound on the ground- state energies of hydrogenic atoms in magnetic fields of arbitrarily large intensity, $E_0$>λ(3t-$t^{\mathrm{-}1}$-2.00232) Ry, where the magnetic field enters through the parameter λ≡B/(4.701×${10}^9$ G) and t is defined implicitly by t(1-$t^2$${)}^{\mathrm{-}2}$=λ$Z^{\mathrm{-}2}$ for atomic number Z. With straightforward modifications, the method presented here can also be employed to derive rigorous analytical lower bounds on the ground-state energies of other parameter-dependent nonseparable quantum systems.

It is pointed out that the approach used by Su to obtain the simplified solution to the Dirac-Coulomb equation can be interpreted in the light of the Infeld-Hull factorization method. According to this method the first-order Dirac-Coulomb equation, after the transformation S, necessarily implies two second-order ‘‘Schrödinger-type’’ equations, whose solutions are the upper and lower simplified solutions of the Dirac-Coulomb equation. It is also interesting to note that the ‘‘recurrence relation’’ between the upper and lower solutions, first obtained by Biedenharn, is exactly the Dirac-Coulomb equation itself, after the transformation S.

The total cross section for electron scattering from CO in the energy range 0.5–5 eV has been measured with use of a time-of-flight spectrometer. This energy region encompasses the ${}_{}{}^2{}_{}{}^{}\mathrm{\pi }$ shape resonance, and a comparison is made with other experimental and theoretical results with regard to the magnitude and position of this structure.

For inelastic scattering of electrons by several representative atomic systems, we test the validity of various separable expansions for the exchange kernel in terms of simple analytical basis functions such as Gaussian-type orbitals (GTO’s). We find that bases of the order of from 10 to 20 GTO functions centered at the nucleus give an accurate representation of the nonlocal exchange terms and produce cross sections in excellent agreement with those from a traditional treatment of exchange.

Vibrational excitation cross sections for ${\mathrm{F}}_2$ have been measured at the impact energy of 0.55–1.75 eV. The vibrational excitation process is proved to be resonant scattering via ${\mathrm{F}}_2$${\mathrm{}}^{\mathrm{-}}$${(}^2$${\mathrm{\Sigma }}_{\mathrm{u}}$). The obtained integrated cross section for v=0→v=1 excitation at $E_i$=1 eV is 1×${10}^{\mathrm{-}16}$ ${\mathrm{cm}}^2$, which is about 1 order of magnitude larger than the value given by the recent semiempirical calculation.

Our earlier measurements of K-shell photoelectric cross sections for intermediate Z elements at 74 and 37 keV have been extended to 26 keV using external conversion x rays in Sn. The experimental results are found to show fairly good agreement with the theoretical values of Scofield.

The method of expansion of the total state in terms of the products of the atomic eigenstates and photon states is used to study the dependence of laser-assisted charge-transfer probabilities and cross sections upon the polarization of radiation. Calculations are made for electron capture into the ground state of hydrogen in ${\mathrm{H}}^{+}$-He collisions in the presence of radiation polarized perpendicular to the nuclear relative velocity. The results are discussed in the light of earlier work for radiation polarized parallel to the relative velocity.

The technique of optical pumping dips has been employed to generate subnatural structure in a ${}_{}{}^{87}{}_{}{}^{}\mathrm{Rb}$ $D_1$ line (natural width 5.6 MHz). Structure as narrow as 1.4 MHz was observed. These experiments represent the first observation of subnatural spectral structure generated without requiring atomic coherence. Additionally, line shapes observed in these studies may be analyzed to yield transition dipole moments without requiring internal standards or an absolute calibration of fluorescence collection efficiency in the experimental apparatus.

The ionization, excitation, and radiative recombination rates for highly stripped ions are cal- culated with Maxwellian and non-Maxwellian electron distribution functions of the type $C_m$exp(-$v^m$/$w^m$) (2≤m≤5) as encountered in laser-heated plasmas and certain types of turbulent plasmas. The direct-radiative-recombination rates are found to decrease by at most 30% as m is increased from 2 to 5. On the other hand, the ionization and excitation rates are found to be strongly reduced if the transition energy χ exceeds twice the local temperature $k_B$$T_e$. The effect of this on the distribution of energy levels and degrees of ionization in laser-produced plasmas could be important.

A dispersion relation which governs the modulational instability of electron cyclotron waves in plasmas is derived. Modulations by pure electronic, ion-acoustic, and quasistatic motion are considered. The growth rates are derived and compared.

A simple experiment on tagged-particle diffusion in a dilute gas has been performed and analyzed. Two lasers were used to prepare and probe a pencil of excited ${}_{}{}^3{}_{}{}^{}\mathrm{P}_1^{}{}_{}{}^{}$ calcium atoms in a background of argon atoms. It was possible to observe the evolution of the ‘‘tagged’’ particles on a time scale extending from free streaming to hydrodynamic (diffusion) behavior. Although this first experiment is only exploratory, a simple analysis of the data indicates the usefulness of the technique for future studies.

The basic features of the notion of charged particles in an axisymmetric mirror, when the magnetic field varies sinusoidally along lines of force, is examined and the diffusion rate for the magnetic moment near the loss cone is obtained.

Oscillations of effective viscosity with density have been observed for helium, argon, nitrogen, and carbon dioxide in the transition regime. The experiments were performed with a torsion pendulum viscometer which has unequal distances between the moving surface and the two fixed surfaces. In general, two predominant oscillations were detected for each gas using Fourier analysis. From the mean positions of these two peaks we have calculated a molecular diameter using a simple relation between the wave lengths in density and the effective molecular cross section. Our values are in reasonable agreement with those measured by other methods for the four gases investigated.

A simple convergent and analytical solution valid for the whole time range for the stochastic Verhulst equation is obtained by treating the strength of the external Gaussian white noise as a perturbation without using the existing approach where the moments are expressed by means of an integral representation. In the case when the deterministic behavior is coupled weakly with the stochastic one induced by the perturbation, it is found that the analytical result is in good agreement with the numerical one. This perturbation technique is also used for treating a class of multiplicative nonlinear stochastic differential equations, and the moments valid for the whole time range are calculated analytically. The result shows explicitly the behavior of the long-time decay of $t^{\mathrm{-}1/2}$, and it also compares satisfactorily with the numerical one in the limit of weak noise strength.

A nonlinear spatially homogeneous Boltzmann equation incorporating both elastic scattering and removal collisions is investigated. The corresponding system of moment equations is solved analytically, and we present the most extended class of similarity solutions for the distribution function f(v,t), which comes out to depend on two time scales and generalizes the solution constructed by Krook and Wu.

Maser action on the hydrogen hyperfine transition has been achieved at a temperature of 0.3 K in a liquid-helium-lined resonator. Polarized atoms are provided by magnetic ejection from a low-temperature discharge in a high magnetic field. The maser demonstrates the feasibility of a proposed low-temperature frequency standard. Operation of the atomic hydrogen source represents an essential step toward magnetic trapping of hydrogen at very low temperatures. Operated continuously, the source provides 5×${10}^{12}$ atoms/sec to the maser. In a 1 sec single pulse it can provide up to 2×${10}^{13}$ polarized atoms.

We report on the design and initial testing of a new type of hydrogen maser, operated at dilution refrigerator temperatures, in which H atoms circulate back and forth between a microwave-pumped state selector and the maser cavity. Other novel design features include liquid${\mathrm{-}}^4$He-coated walls, He-cooled electronics, and the use of microscopic magnetic particles to relax the two lowest hyperfine levels in the state selector. Stabilities at least as good as that of a Rb clock and a high-stability quartz oscillator are observed for measuring times between 1 and 300 s.

We point out that hole burning offers a test of newly proposed theories of generalized resonance relaxation. In contrast to the usual Bloch theory of relaxation, the generalized theory has off-diagonal and intensity-dependent damping coefficients. We show that one observable consequence of our theory is a narrowed non-Lorentzian hole.